Harm osc, anharmonic perturbation

In summary, the mean position of a particle bound by the potential V = \frac{1}{2} m \omega^2 x^2 - a x^3 changes with the energy of the eigenstates when the coefficient a is small, allowing for first order perturbation theory to be used. This is shown by calculating the mean position using the expansion of eigenkets and taking the inner product with the position operator. This approach is preferred over calculating the perturbated energy to second order.
  • #1
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Homework Statement


Particle bound by
[tex] V = \frac{1}{2} m \omega^2 x^2 - a x^3 [/tex]
for small x. Show that the mean position of the particle changes with the energy of the eigenstates when [tex]a[/tex] is small, so first order perturbation theory works.

Homework Equations


For the harmonic oscillator
[tex] x = \sqrt{\frac{\hbar}{2m\omega}}(a^{\dagger}+a) [/tex]


The Attempt at a Solution


That x^3 perturbation will give an odd number of creation/destruction operators, so there's no shift in energy eigenvalues to first order in perturbation theory. But how does that help answering the question?
 
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  • #2
There is a first order transition, but it is between different energy states.
 
  • #3
But how does that help answering the question?
 
  • #4
Noone have any ideas on this?
 
  • #5
I'm not going to give up on this! The question asks to calculate the mean position of the particle, so if I do that:
[tex]<x> = \sqrt{\frac{\hbar}{2m\omega}}<n|a^{\dagger}+a|n> = 0 [/tex]
That's just zero, haveing nothing to do with energy eigernstates or size of a?

Sure I could calculate the perturbated energy to second order, getting terms like
[tex] a^2 \big( \frac{\hbar}{2m\omega} \big)^3 <n|a^{\dagger} aa \frac{1}{E_n-H_0} a^{\dagger} a a^{\dagger} |n> = -n(n+1)^2 \frac{\hbar^2 a^2}{8m^3 \omega^4}[/tex]
and so on, but I don't see how that has anyhting to do with the question asked.
 
  • #6
Shouldn't you consider the expanded eigenkets?

[tex] | n > = |n^{(0)}>+\lambda |n^{(1)}> + \lambda^2 |n^{(2)}> + \dots[/tex]

Then you get to first order something like:

[tex] <x> = < n^{(0)} | x | n^{(0)} > + \lambda \left( < n^{(0)} | x | n^{(1)} > + < n^{(1)} | x | n^{(0)} > \right)+ \mathcal{O} (\lambda^2) = \lambda \left( < n^{(0)} | x | n^{(1)} > + < n^{(1)} | x | n^{(0)} > \right)+ \mathcal{O} (\lambda^2)[/tex]

Could that be the correct approach?
 
Last edited:
  • #7
I think that you don't really have to solve for the corrected energy's. You need the new eigenkets, and then just solve for the expectation value, <x>, by taking the inner product of x and the eigenkets.
 

1. What is Harm osc, anharmonic perturbation?

Harm osc, anharmonic perturbation is a mathematical model used in physics to describe the behavior of a system that is subject to an external perturbation or disturbance. It takes into account the effects of anharmonicity, which means that the system does not follow a simple harmonic motion like a pendulum, for example.

2. How is Harm osc, anharmonic perturbation used in scientific research?

Harm osc, anharmonic perturbation is used in various fields of physics, such as solid-state physics, quantum mechanics, and statistical mechanics. It allows scientists to analyze and understand the behavior of complex systems that cannot be described by simple harmonic motion.

3. What are the equations used in Harm osc, anharmonic perturbation?

The equations used in Harm osc, anharmonic perturbation depend on the specific system being studied. In general, they involve a combination of equations for harmonic motion and an additional term for anharmonicity. These equations can be solved numerically or approximated using perturbation theory.

4. How does Harm osc, anharmonic perturbation differ from simple harmonic motion?

Simple harmonic motion describes the behavior of a system that oscillates back and forth with a constant amplitude and frequency. In contrast, Harm osc, anharmonic perturbation takes into account the effects of nonlinearity and anharmonicity, which can cause the amplitude and frequency of the oscillations to change over time.

5. What real-world applications does Harm osc, anharmonic perturbation have?

Harm osc, anharmonic perturbation has various applications in physics, such as in the study of crystal structures, molecular vibrations, and the behavior of nanoparticles. It is also used in engineering to understand and design complex systems, such as bridge structures and electrical circuits.

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